Exact solutions for nonlinear partial differential equations via a fusion of classical methods and innovative approaches

This paper presents a new approach for finding exact solutions to certain classes of nonlinear partial differential equations (NLPDEs) by combining the variation of parameters method with classical techniques such as the method of characteristics. Our primary focus is on NLPDEs of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{tt}+a(x,t)u_{xt}+b(t)u_{t}=\alpha (x,t)+ G(u)(u_{t}+a(x,t)u_{x})e^{-\int b(t)dt}$$\end{document}utt+a(x,t)uxt+b(t)ut=α(x,t)+G(u)(ut+a(x,t)ux)e-∫b(t)dt and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{t}^{m}(u_{tt}+a(x,t)u_{xt})+b(t)u_{t}^{m+1}=e^{-(m+1)\int b(t)dt}(u_{t}+a(x,t)u_{x}) F(u,u_{t}e^{\int b(t)dt}).$$\end{document}utm(utt+a(x,t)uxt)+b(t)utm+1=e-(m+1)∫b(t)dt(ut+a(x,t)ux)F(u,ute∫b(t)dt). We provide numerical validation through several examples to ensure accuracy and reliability. Our approach enhances the applicability of analytical solution methods for a broader range of NLPDEs.

conservation laws.The exact procedure for this reduction may vary depending on the specific equation and the desired format for further analysis.
However, the method of characteristics and the variation of parameters are two distinct methods used in different contexts.While these two methods have distinct applications, this study shows that combining the classical techniques derives new solutions for NLPDEs with specific initial conditions.
Several analytical methods consistently solve classes of second-order differential equations by variation of parameters.In 31 and 33 , some types of nonlinear differential equations have been reduced to first-order using suitable parameter variations.The resulting first-order differential equations are, in most cases, transformable to well-known integrable or solvable classical differential equations.However, these methods are not applicable when dealing with nonlinear partial differential equations.Certain types of NLPDEs remain unsolvable using variations of parameters independently.By leveraging the strengths of classical techniques, we demonstrate an expanded scope of solvable NLPDEs, thus increasing the applicability of analytical solution methods to a broader range of problems.
As an extension of a previous study 36 , we introduced new solutions to NLPDEs.In this study, we consider the classes of nonlinear partial differential equations of the form: and Notably, some exceptional cases can arise.For example, we mention the nonlinear differential equations recorded in 31 , where the functions were restricted to one variable.
The remainder of this paper is organized as follows.In "First class of reducible nonlinear partial differential equations", we apply our methodology to the first class of reducible second-order partial differential equations to determine the exact solutions of NLPDEs of the first type.
"Second class of reducible nonlinear partial differential equations" delves into the second class of reducible nonlinear partial differentiable equations.Based on these results, a new class of solutions was derived.We demonstrate the application of the proposed method using concrete examples to demonstrate its viability and efficiency.Using Mathematica algorithms, relevant numerical representations were exhibited in each example to show the pertinence of obtained analytical solutions.Finally, "Conclusion" concludes the paper.

Description of the method and construction of the general solutions
We consider the first class of nonlinear second-order partial differential equations compilable in the following general form: where u denotes a function of (x, t) ∈ R 2 .
First, we solve the characteristic equation Then (1) can be rewritten as Multiplying both sides of ( 2) by e b(t)dt , we get The nonlinear second-order partial differential equation (1) can be solved easily if we assume that where H and K are differentiable functions of t and u respectively.Then, we differentiate to obtain Substituting (4) into (3), we find that: and (1) (2) (5) Remark 1 Let G = u n , where n is a non zero positive integer.
Then, the second order partial differential equation (1) becomes Applying Eqs. ( 5) and ( 6), we get an Abel equation of the form A comprehensive compilation of integrable Abel equations can be found in [37][38][39] and 40 . Application with the initial conditions u(x, 0) = 1 and u t (x, 0) = x + 1.

Solution:
We solve the characteristic equation which leads to x(t) = x 0 e t .The functions H and K are general solutions of Then we get and where C 1 and C 2 are arbitrary constants.
The second-order partial differential equation ( 7) is reduced to the first-order differential equation The first-order differential equation ( 8) is an Abel equation, which can be solved using various methods.For more details, refer to [37][38][39] .
Using initial condition u(x, 0) = 1 and u t (x, 0) = x + 1 , we obtain explicit solutions of (7) Visualizing the precise solutions obtained by Mathematica algorithms (Fig. 1) and plotting solution profiles at different values of t, we observe the characteristics of several solutions of ( 7) with initial conditions u(x, 0) = 1 and u t (x, 0) = x + 1 .As a result, these solutions develop singularities at certain values of x and t.Note that despite the smoothness of the initial data, the spontaneous singular behavior in the solutions must be due to the nonlinear term of the equation.Figure 1 displays the 2D, 3D and contour plots of the solutions in (7) within −10 ≤ x ≤ 10 and 0 ≤ t ≤ 4 for 3D and contour graphs, t = 1 for 2D graph.

Second class of reducible nonlinear partial differential equations Description of the method and construction of the general solutions
The second group of second-order partial differential equations is formulated as follows: As in the previous section, the second-order nonlinear partial differential equation ( 9) can be readily solved if we suppose that where u = u(x(t), t) and K is a differentiable function.Then Differentiate (11) to obtain Then Substituting Eqs. ( 10), ( 11) and ( 12) into (13), we find Therefore, the following statement holds.(10)

Solution:
Using the previous result, we find that the second-order nonlinear partial differential equation can be reduced to the first-order differential equation where K is the general solution of K ′ (u) = K(u).Taking x(t) = t + x 0 , we obtain K(u) = A(x 0 )e u , where A is an arbitrary constant of integration.The differential equation ( 15) takes the form and the exact solutions of the second-order nonlinear partial differential equation ( 14) are analytically determined and take the following form: where F and G are arbitrary functions.It follows from the initial conditions at (x 0 , 0) given by u(x, 0) = 0 and u t (x, 0) = x 2 that the exact solutions of ( 14) can be expressed explicitly as follows Envisioning the precise solutions obtained by Mathematica (Fig. 2) and plotting solution profiles at different values of t, we have seen equations with smooth coefficients and initial data develop spontaneous singularities due to the nonlinearity of the equations.The solutions of ( 14) break down at some values of x and t, and no classical solution for the initial value problems exists beyond this point of breakdown.
Note that the nonlinear partial differential Eq. ( 14) yields a more straightforward solution than the initial value problem in the previous example.
Suppose the function F satisfies F(s, w) = w.

Solution:
Using the previous result, we find that the second-order nonlinear partial differential equation can be reduced to the first-order differential equation where K is the general solution of K ′ (u) = K(u).
The differential equation ( 17) takes the form and the exact solutions of the second order nonlinear partial differential equation ( 16) are analytically determined and take the following form: where F and G are arbitrary functions.
It follows from the initial conditions at (x 0 , 0) given by u(x, 0) = 0 and u t (x, 0) = x 2 that the exact solutions of ( 16) can be expressed explicitly as follows When we envision the exact solutions of ( 16) generated by Mathematica as depicted in Fig. 3 and create plots showing the solution profiles at various time points, we find that they deteriorate at specific values of both x and t.Beyond this point, a classical solution is no longer viable for the initial value problems.

Solution:
Using the previous result, we find that the second-order nonlinear partial differential equation can be reduced to the first-order differential equation where K is the general solution of K ′ (u) = u.
The differential equation ( 19) takes the form where f is an arbitrary function that leads to a Ricatti differential equation.
Taking in account the initial conditions u(x, 0) = 0 and u t (x, 0) = x 2 , the differential equation (19) becomes The result was obtained using Mathematica code as a complicated function.As in the previous examples, the nonlinearity of the partial differential equations produces singular behavior in the solutions.Figure 4 shows the 2D, 3D and contour plots of the solutions in (18) within −2 ≤ x ≤ 2 and 0 ≤ t ≤ 2 for 3D and contour graphs, t = 2 for 2D graph.

Solution:
Using the previous result, we find that the second-order nonlinear partial differential equation can be reduced to the first-order differential equation where K is the general solution of K ′ (u) = u 2 .Differential equation (21) takes the form of an Abel equation (20) where K is the general solution of K ′ (u) = 2 u K(u) + 1 u 2 K 2 (u).We get For a = 1 , (22) takes the form Then the general solutions of ( 25) are given by where f denotes an arbitrary function.
Remark 3 Let u(x, t) = t −1 B(x) be a family of solutions to Eq. (25).
If f (x, t) = A , we get which is an implicit solution of (25) where A an arbitrary constant and B is a function of x.
We checked the implicit solutions of ( 25) by generating Mathematica codes and considering the initial conditions u(x, 1) = 1 and u t (x, 1) = x 2 .
(24) u t (x, t)) = K(u) 1 t ,  with the initial value conditions u(x, 1) = x and u t (x, 1) Remark 4 In 31 , the author studied special case of (26) where u is a single variable function of t.In ( 26), if we suppose that u is only a function of t, we get which is the differential equation (61) investigated by the authors in 31 .

Solution:
(26) is reduced to the differential equation where K is the general solution of Then the general solutions of 26 are given by ( 26) www.nature.com/scientificreports/where f is an arbitrary function.By generating Mathematica codes, we obtain implicit solutions of (26) Figure 7 shows the 2D, 3D and contour plots of the solutions in (26) within −1 ≤ x ≤ 1 and 0 ≤ t ≤ 2 for 3D and contour graphs, t = 2 for 2D graph.
Remark 5 A special case of our findings was recorded in 31 .If we suppose that u is only a function of t in (28), we get which is exactly the differential equation (78) investigated by the authors of 31 .

Solution:
(28) is reduced to the differential equation where K satisfies Two particular solutions to (30) are given by K 1 (u) = 2u and K 2 (u) = −u.